Fleeing from justice is an overall bad idea. Not only will you be considered a fugitive, but there are no legal benefits for trying to escape criminal prosecution.
Of course, that hasn't stopped criminals from trying to hide from law enforcement. But even some of the most notorious fugitives like mob boss Whitey Bulger eventually get caught. Same with lesser-known folks who try to get away with things like being a deadbeat parent.
So while each situation is different, here are a few examples of what can happen to criminals who try to flee from justice:
The statute of limitations for the crime won't apply. For federal crimes, the statute of limitations for the crime committed doesn't run out just because the criminal is on the run. For example, if the statute of limitations for a crime is five years, but the criminal intentionally hides out in another country for 10 years to avoid prosecution, he can still be prosecuted for the crime if he's caught even though the five-year limitations period has passed.
Your mugshot may make you (in)famous. Criminals attempting to escape the law are not only facing legal problems, but could also face reputational issues. Law enforcement frequently shares mugshots of wanted criminals on its social media pages and with the media, so everyone will know you're fleeing from justice.
Bounty hunters can potentially track you down. Bounty hunters are authorized by the law to track down criminal defendants who skip out on their bail and fail to appear for their court dates. Bounty hunters have a monetary incentive to track down criminals on the run, so even if law enforcement officers don't catch you, a bounty hunter might.
You may face extradition. Criminals who flee to another state or country can face extradition back to the jurisdiction where the crime occurred. For criminals who've fled to other states, the state seeking extradition must file the proper documents, show that you've been charged with a crime in that state, and show that you're a fugitive. This means that you intentionally went to another jurisdiction to avoid getting caught.